Spectral Graph TheorySpectral Graph Theory Social and Technological Networks Rik Sarkar University...
Transcript of Spectral Graph TheorySpectral Graph Theory Social and Technological Networks Rik Sarkar University...
SpectralGraphTheory
SocialandTechnologicalNetworks
Rik Sarkar
UniversityofEdinburgh,2018.
Spectralmethods
• Understandingagraphusingeigen valuesandeigen vectorsofthematrix
• Wesaw:• Ranksofwebpages:componentsof1steigenvectorofsuitablematrix
• Pagerank orHITSarealgorithmsdesignedtocomputetheeigen vector
• Randomwalksandlocalpageranks helpinunderstandingcommunitystructure
Laplacian
• L=D– A[Disthediagonalmatrixofdegrees]
• Aneigen vectorhasonevalueforeachnode• Weareinterestedinpropertiesofthesevalues
2
664
1 �1 0 0�1 2 �1 00 �1 2 �10 0 �1 1
3
775 =
2
664
1 0 0 00 2 0 00 0 2 00 0 0 1
3
775�
2
664
0 1 0 01 0 1 00 1 0 10 0 1 0
3
775
Laplacian
• L=D– A[Disthediagonalmatrixofdegrees]
• Symmetric.RealEigenvalues.• Rowsum=0.Singularmatrix.Atleastoneeigenvalue=0.
• Positivesemidefinite.Non-negativeeigen values
2
664
1 �1 0 0�1 2 �1 00 �1 2 �10 0 �1 1
3
775 =
2
664
1 0 0 00 2 0 00 0 2 00 0 0 1
3
775�
2
664
0 1 0 01 0 1 00 1 0 10 0 1 0
3
775
Laplacian andrandomwalks
• Supposewearedoingarandomwalkonagraph• Letu(i)betheprobabilityofthewalkbeingatnodei– E.g.initiallyitisatstartingnodes– After10steps,probabilityishighernears,lowatnodesfartheraway
– Question:Howdoestheprobabilitychangewithtime?
– Thisprobabilitydiffuseswithtime.Likeheatdiffuses
Laplacian matrix
• Imagineasmallanddifferentquantityofheatateachnode(say,inametalmesh)
• wewriteafunctionu:u(i)=heatati• Thisheatwillspreadthroughthemesh/graph• Question:howmuchheatwilleachnodehaveafterasmallamountoftime?
Heat diffusion
• Supposenodesi andjareneighbors– Howmuchheatwillflowfromi toj?
Heat diffusion
• Supposenodesi andjareneighbors• Inashorttime,howmuchheatwillflowfromi toj?
• Proportionaltothegradient:(u(i)- u(j))*∆𝑡– Letuskeep∆𝑡fixed,andwritejust(u(i)- u(j))
• thisissigned:negativemeansheatflowsintoi
Heat diffusion
• Ifi hasneighborsj1,j2….• Thenheatflowingoutofi is:
=(u(i)- u(j1))+(u(i)- u(j2))+(u(i)- u(j3))+…=degree(i)*u(i)- u(j1)- u(j2)- u(j3)- ….
• HenceL=D- A
The heat equation
• Thenetheatoutflowofnodesinatimestep• Thechangeinheatdistributioninasmalltimestep– Therateofchangeofheatdistribution
@u
@t= L(u)
Thesmoothheatequation
• ThesmoothLaplacian:
• Thesmoothheatequation:
�f =@f
@t
Heatflow
• Willeventuallyconvergetov[0]:thezeroth eigenvector,witheigen value�0 = 0
v[0]=const forthe chain
Eigenvectors
• Othereigen vectors• Encodevariouspropertiesofthegraph• Havemanyapplications
Application1:Drawingagraph(Embedding)
• Problem:Computerdoesnotknowwhatagraphissupposedtolooklike
• Agraphisajumbleofedges
• Consideragridgraph:• Wewantitdrawnnicely
Graphembedding
• Findpositionsforverticesofagraphinlowdimension(comparedton)
• Commonobjective:Preservesomepropertiesofthegraphe.g.approximatedistancesbetweenvertices.Createametric– Usefulinvisualization– Findingapproximatedistances– Clustering
• Usingeigen vectors– Oneeigen vectorgivesxvaluesofnodes– Othergivesy-valuesofnodes…etc
Drawwithv[1]andv[2]
• Supposev[0],v[1],v[2]…areeigenvectors– Sortedbyincreasingeigenvalues
• PlotgraphusingX=v[1],Y=v[2]
• Producesthegrid
Intuitions:the1-Dcase
• Supposewetakethejth eigen vectorofachain
• Whatwouldthatlooklike?• Wearegoingtoplotthechainalongx-axis• Theyaxiswillhavethevalueofthenodeinthejth eigen vector
• Wewanttoseehowtheseriseandfall
Observations
• j=0
• j=1
• j=2
• j=3
• j=19
For Allj
• Lowonesatbottom
• Highonesattop
• Codeonwebpage
Observations
• InDim 1grid:– v[1]ismonotone– v[2]isnotmonotone
• Indim2grid:– bothv[1]andv[2]aremonotoneinsuitabledirections
• Forlowvaluesofj:– Nearbynodeshavesimilarvalues• Usefulforembedding
Application2:Colouring
• Colouring:Assigncolours tovertices,suchthatneighboringverticesdonothavesamecolour– E.g.Assignmentofradiochannelstowirelessnodes.Goodcolouring reducesinterference
• Idea:Higheigen vectorsgivedissimilar valuestonearbynodes
• Useforcolouring!
Application3:Cuts/segmentation/clustering
• Findthesmallest‘cut’• Asmallsetofedgeswhoseremovaldisconnectsthegraph
• Clustering,communitydetection…
Clustering/communitydetection
• v[1]tendstostretchthenarrowconnections:discriminatesdifferentcommunities
Clustering:communitydetection
• Morecommunities• Spectralembeddingneedshigherdimensions
• Warning:itdoesnotalwaysworksocleanly
• Inthiscase,thedataisverysymmetric
ImagesegmentationShi&malik’00
Laplacian
• ChangedimpliedbyLonanyinputvectorcanberepresentedbysumofactionofitseigenvectors(wesawthislasttimeforMMT)
• v[0]istheslowestcomponentofthechange– Withmultiplierλ0=0– Thesteadystatecomponent
• v[1]isslowestnon-zerocomponent– withmultiplierλ1
Spectralgap• λ1– λ0
• Determinestheoverallspeedofchange• Iftheslowestcomponentv[1]changesfast– Thenoverallthevaluesmustbechangingfast– Fastdiffusion
• If theslowest componentis slow– Convergencewill beslow
• Examples:– Expanders have largespectral gaps– Grids anddumbbellshave smallgaps~1/n
Application4:isomorphismtesting
• Eigenvaluesbeingdifferentimpliesgraphsaredifferent
• Thoughnotnecessarilytheotherway
Spectralmethods• Wideapplicabilityinsideandoutsidenetworks• Relatedtomanyfundamentalconcepts
– PCA– SVD
• Randomwalks,diffusion,heatequation…• Resultsaregoodmanytimes,butnotalways• Relativelyhardtoproveandunderstandproperties• Inefficient:eig.computationcostlyonlargematrix• (Somewhat)efficientmethodsexistformorerestricted
problems– e.g.whenwewantonlyafewsmallest/largesteigen vectors